HAL Id: hal-00530175
https://hal.archives-ouvertes.fr/hal-00530175
Submitted on 27 Oct 2010
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
A Nonlinear Adiabatic Theorem for Coherent States
Rémi Carles, Clotilde Fermanian Kammerer
To cite this version:
Rémi Carles, Clotilde Fermanian Kammerer. A Nonlinear Adiabatic Theorem for Coherent States.
Nonlinearity, IOP Publishing, 2011, 24 (8), pp.2143-2164. �10.1088/0951-7715/24/8/002�. �hal-
00530175�
STATES
R´EMI CARLES AND CLOTILDE FERMANIAN-KAMMERER
Abstract. We consider the propagation of wave packets for a one-dimensional nonlinear Schr¨odinger equation with a matrix-valued potential, in the semi- classical limit. For an initial coherent state polarized along some eigenvector, we prove that the nonlinear evolution preserves the separation of modes, in a scaling such that nonlinear effects are critical (the envelope equation is non- linear). The proof relies on a fine geometric analysis of the role of spectral projectors, which is compatible with the treatment of nonlinearities. We also prove a nonlinear superposition principle for these adiabatic wave packets.
1. Introduction
We consider the semi-classical limit ε → 0 for the nonlinear Schr¨odinger equation (1.1)
iε∂
tψ
ε+ ε
22 ∂
x2ψ
ε= V (x)ψ
ε+ Λ | ψ
ε|
2CNψ
ε, (t, x) ∈ R × R, ψ
|t=0ε= ψ
0εwhere Λ ∈ R. The data ψ
0εand the solution ψ
ε(t) are vectors of C
N, N > 1. The quantity | ψ
ε|
2CNdenotes the square of the Hermitian norm in C
Nof the vector ψ
ε. Finally, the potential V is smooth and valued in the set of N by N Hermitian matrices. Such systems appear in the modelling of Bose-Einstein condensate (see [1]
and references therein).
Definition 1.1. We say that a function f is at most quadratic if f ∈ C
∞(R) and for all k > 2, f
(k)∈ L
∞(R).
We make the following assumptions on the potential V :
Assumption 1.2. (1) We have V (x) = D(x)+W (x) with D, W ∈ C
∞(R, R
N×N), D diagonal with at most quadratic coefficients, and W symmetric and bounded as well as its derivatives, W ∈ W
∞,∞(R).
(2) The matrix V has P distinct, at most quadratic, eigenvalues λ
1, . . . , λ
Pand (1.2) ∃ c
0, n
0∈ R
+, ∀ j 6 = k, ∀ x ∈ R , | λ
j(x) − λ
k(x) | > c
0h x i
−n0.
Under these assumptions (the first point suffices), we can prove global existence of the solution ψ
εfor fixed ε > 0:
Lemma 1.3. If V satisfies Assumption 1.2 and ψ
0ε∈ L
2( R ), there exists a unique, global, solution to (1.1)
ψ
ε∈ C R; L
2(R)
∩ L
8locR; L
4(R) .
This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01).
1
The L
2-norm of ψ
εdoes not depend on time: k ψ
ε(t) k
L2(R)= k ψ
ε0k
L2(R), ∀ t ∈ R.
The proof of this lemma is sketched in Appendix A.
In this nonlinear setting, the size of the initial data is crucial. As in [4], we choose to consider initial data of order 1 (in L
2), and to introduce a dependence upon ε in the coupling constant (note that the nonlinearity is homogeneous). This leads to the equation
iε∂
tψ
ε+ ε
22 ∂
x2ψ
ε= V (x)ψ
ε+ Λε
2β| ψ
ε|
2CNψ
ε,
and we choose the exponent β = 3/4, which is critical for the type of initial data we want to consider (coherent state) when the potential V is scalar (see [4]). We are left with the nonlinear semi-classical Schr¨odinger equation
(1.3) iε∂
tψ
ε+ ε
22 ∂
x2ψ
ε= V (x)ψ
ε+ Λε
3/2| ψ
ε|
2CNψ
ε; ψ
|t=0ε= ψ
ε0. We focus on initial data which are perturbation of wave packets
(1.4) ψ
0ε(x) = ε
−1/4e
iξ0(x−x0)/εa
x − x
0√ ε
χ(x) + r
0ε(x), where the initial error satisfies
(1.5) k r
ε0k
L2(R)+ k xr
0εk
L2(R)+ k ε∂
xr
ε0k
L2(R)= O (ε
κ) for some κ > 1 4 . The profile a belongs to the Schwartz class, a ∈ S (R; C), and the initial datum is polarized along an eigenvector χ(x) ∈ C
∞(R; C
N):
V (x)χ(x) = λ
1(x)χ(x), with | χ(x) |
CN= 1.
Note that λ
1is simply a notation for some eigenvalue, up to a renumbering of eigenvalues. The L
2-norm of ψ
ε0is independent of ε, k ψ
0εk
L2(R)= k a k
L2(R). As pointed out above, this is equivalent to considering (1.1) with initial data of the same form (1.4), but of order ε
3/4in L
2(R). The evolution of such data when a is a Gaussian has been extensively studied by G. Hagedorn on the one hand, and by G. Hagedorn and A. Joye on the other hand, in the linear context Λ = 0 (see [8, 9]).
These data are also particularly interesting for numerics (see [12] and the references therein).
Because of the gap condition, the matrix V has smooth eigenvalues and eigenpro- jectors (see [11]). Besides, the gap condition (1.2) also implies that we control the growth of the eigenprojectors (see Lemma C.2). Note however than in dimension 1 (x ∈ R ), one can have smooth eigenprojectors without any gap condition. We explain this fact below and give an example of projectors that we can consider; we also illustrate why things may be more complicated in higher dimensions (d > 2).
Example 1.4. For N = 2 and x ∈ R, consider (1.6) V (x) = (ax
2+ b)Id +
u(x) v(x) v(x) − u(x)
,
for a, b ∈ R, and u and v smooth and bounded with bounded derivatives. Such a potential satisfies Assumption 1.2. Its eigenvalues are the two functions
λ
±(x) = ax
2+ b ± p
u(x)
2+ v(x)
2.
These functions are clearly smooth outside the set of points x
0such that u(x
0)
2+
v(x
0)
2= 0. Besides, for such points, one can renumber the modes in order to
build smooth eigenvalues. More precisely, observe first that if u(x)
2+ v(x)
2= O((x − x
0)
∞) close to x
0, the functions λ
±are smooth close to x
0. Moreover, if u(x)
2+ v(x)
2= (x − x
0)
kf (x) with f (x
0) 6 = 0, necessarily f (x
0) > 0 and k = 2p, so we have
λ
±(x) = ax
2+ b ± | x − x
0|
pp f (x).
For p even these functions are again smooth. However, when p is odd, they are no longer smooth and we perform a renumbering of the eigenfunctions, observing that
x 7→ ax
2+ b + (x − x
0)
pp f (x) are smooth eigenvalues of V close to x
0.
Example 1.5. Resume the above example, with now x ∈ R
d, d > 2. The smoothness of the eigenvalues is no longer guaranteed: suppose u(x) = x
1and v(x) = x
2, then the functions λ
±are not smooth and one cannot find any renumbering which makes them smooth.
Example 1.6. For an example of a potential which satisfies 1.2, we simply choose V as in (1.6) with
c
uu(x) = c
vv(x) = h x i
−n0, c
2u+ c
2v6 = 0.
1.1. The ansatz. We consider the classical trajectories (x(t), ξ(t)) solutions to (1.7) x(t) = ˙ ξ(t), ξ(t) = ˙ −∇ λ
1(x(t)), x(0) = x
0, ξ(0) = ξ
0.
Because λ
1is at most quadratic, the classical trajectories grow at most exponen- tially in time (see e.g. [4]):
(1.8) ∃ C > 0, | ξ(t) | + | x(t) | . e
Ct. We denote by S the action associated with (x(t), ξ(t))
(1.9) S(t) =
Z
t 01
2 | ξ(s) |
2− λ
1(x(s))
ds.
We consider the function u = u(t, y) solution to (1.10) i∂
tu + 1
2 ∂
y2u = 1
2 λ
′′1(x(t)) y
2u + Λ | u |
2u ; u(0, y) = a(y), and we denote by ϕ
εthe function associated with u, x, ξ, S by:
(1.11) ϕ
ε(t, x) = ε
−1/4u
t, x − x(t)
√ ε
e
i(S(t)+ξ(t)(x−x(t)))/ε.
Global existence of u and control of its derivatives and momenta are proved in [3].
More precisely, we have the following result.
Theorem 1.7 (From [3]). Suppose a ∈ S (R). There exists a unique, global so- lution u ∈ C (R; L
2(R)) ∩ L
8loc(R; L
4(R)) to (1.10). In addition, for all k, p ∈ N, h y i
k∂
ypu ∈ C ( R ; L
2( R )) and
(1.12) ∀ k, p ∈ N, ∃ C > 0, ∀ t ∈ R
+, k h y i
k∂
ypu(t, · ) k
L2(R). e
Ct.
In particular, note that ∂
pyu(t, · ) is in L
∞for all p ∈ N. These results have
consequences on ϕ
ε. As far as the L
∞norm is concerned, we infer, using (1.8),
(1.13) ∀ p ∈ N , k (ε∂
x)
pϕ
ε(t) k
L∞. ε
−1/4e
Cpt.
We use the time-dependent eigenvectors constructed in [8] (see also [9] and [14]).
To make the notations precise, we denote by d
jthe multiplicity of the eigenvalue λ
j, 1 6 j 6 P (note that P
16j6P
d
j= N).
Proposition 1.8. There exists a smooth orthonormal family χ
ℓ(t, x)
16ℓ6d1
such that for all t, χ
ℓ(t, x)
16ℓ6d1
spans the eigenspace associated to λ
1, χ
1(0, x) = χ(x) and for m ∈ { 1, · · · , d
1} ,
(1.14) χ
m(t, x), ∂
tχ
ℓ(t, x) + ξ(t)∂
xχ
ℓ(t, x)
CN
= 0.
Moreover, for ℓ ∈ { 1, . . . , d
1} , k, p ∈ N, there exists a constant C = C(p, k) such that ∂
tp∂
xkχ
ℓ(t, x)
CN6 C e
Cth x i
(k+p)(1+n0),
where n
0appears in (1.2).
Note that equation (1.14) for m = ℓ is true as soon as the eigenvector χ
ℓis normalized and real-valued.
Equation (1.14) is often referred to as parallel transport. These time-dependent eigenvectors are commonly used in adiabatic theory and are connected with the Berry phase (see [14]). Their construction is recalled in Section 2, where the control of their growth is also established.
Notation. In the case of a single coherent state, we complete the family χ
ℓ(t, x)
16ℓ6d1
as an orthonormal basis χ
ℓj16j6P 16ℓ6dj
of C
Nas follows:
• χ
ℓ1= χ
ℓ,
• For j > 2 and 1 6 ℓ 6 d
j, χ
ℓj= χ
ℓj(x) does not depend on time,
• For j > 2, (χ
ℓj)
16j6djspans the eigenspace associated to λ
j.
1.2. The results. We prove that there is adiabatic decoupling for the solution of (1.3) with initial data which are coherent states of the form (1.4): the solution keeps the same form and remains in the same eigenspace.
Theorem 1.9. Let a ∈ S (R) and r
ε0satisfying (1.5). Under Assumption 1.2, con- sider ψ
εsolution to the Cauchy problem (1.3)–(1.4), and the approximate solution ϕ
εgiven by (1.11). There exists a constant C > 0 such that the function
w
ε(t, x) = ψ
ε(t, x) − ϕ
ε(t, x)χ
1(t, x), where χ
1is given by Proposition 1.8, satisfies
sup
|t|6Cloglog1ε
( k w
ε(t) k
L2+ k xw
ε(t) k
L2+ k ε∂
xw
ε(t) k
L2) −→
ε→00.
This adiabatic decoupling between the modes is well-known in the linear setting and is at the basis of numerous results on semi-classical Schr¨odinger operator with matrix-valued potential in the framework of Born-Oppenheimer approximation for molecular dynamics. On this subject, the reader can consult the article of H. Spohn and S. Teufel [13] or the book of S. Teufel [14] for a review on the topic (see also [2]
for an adiabatic result in a nonlinear context and [10] for application of adiabatic theory to the obtention of resolvent estimates).
Remark 1.10. Suppose that V depends on ε with V
ε= D + εW , where D and W are as in Assumption 1.2; this is so in the model presented in [1]. Then the above result remains true for | t | 6 C log
1ε: we gain one logarithm. See Remark 4.1 for
the key arguments. Also, the assumption on the initial error can be relaxed: to prove the analogue of Theorem 1.9 with an approximation in L
2up to C log 1/ε, (1.5) can be replaced with
k r
ε0k
L2(R)→ 0 as ε → 0.
In contrast with the general framework of this paper, no rate is needed: the rate in (1.5) is due to the fact that we cannot use Strichartz estimates here.
It is also interesting to analyze the evolution of solution associated with data which are the superposition of two data of the studied form. We suppose
ψ
0ε(x) = ϕ
ε1(0, x)χ
1(x) + ϕ
ε2(0, x)χ
2(x),
where both functions ϕ
ε1and ϕ
ε2have the form (1.11), for two eigenvectors of V , χ
1and χ
2, and phase space points (x
1, ξ
1) and (x
2, ξ
2). We assume
(χ
1, x
1, ξ
1) 6 = (χ
2, x
2, ξ
2) .
We associate with the phase space points (x
j, ξ
j), j ∈ { 1, 2 } the classical trajectories (x
j(t), ξ
j(t)), and the action S
j(t) associated with ˜ λ
jsuch that
V (x)χ
j(x) = ˜ λ
j(x)χ
j(x).
Note that we may have ˜ λ
1= ˜ λ
2. Let us denote by χ
ℓj(t)
16j6P 16ℓ6dja time-dependent orthonormal basis of eigenvectors defined according to Proposition 2.1 (see also Proposition 1.8 above) with χ
11(0, x) = χ
1(x), χ
2(x) = χ
21(0, x) if ˜ λ
1= ˜ λ
2, χ
2(x) = χ
12(0, x) otherwise, and by ϕ
εjthe ansatz defined by (1.11). To unify the presenta- tion, we write
χ
1= χ
11; χ
2=
( χ
21if ˜ λ
1= ˜ λ
2, χ
12otherwise.
Theorem 1.11. Set E
j=
ξ2 j
2
+ ˜ λ
j(x
j) for j ∈ { 1, 2 } and suppose Γ = inf
x∈R
λ ˜
1(x) − λ ˜
2(x) − (E
1− E
2) > 0.
There exists C > 0 such that the function
w
ε(t) = ψ
ε(t) − ϕ
ε1χ
1(t, x) − ϕ
ε2χ
2(t, x).
satisfies
sup
t6Cloglog1ε
( k w
ε(t) k
L2+ k xw
ε(t) k
L2+ k ε∂
xw
ε(t) k
L2) −→
ε→00.
Note that if ˜ λ
1= ˜ λ
2, one recovers the condition E
16 = E
2of [4]. The proof of Theorem 1.11 follows the same lines as in [4, Section 6]. The constant Γ controls the frequencies of time interval where trajectories cross.
Remark 1.12. In finite time, the situation is different whether ˜ λ
1= ˜ λ
2or not. If λ ˜
1= ˜ λ
2, the superposition holds in finite time without any condition on Γ; this comes from the fact that the trajectories x
1(t) and x
2(t) only cross on isolated points (see [4]). However, if ˜ λ
16 = ˜ λ
2one may have x
1(t) = x
2(t) on intervals of non-empty interior: the condition Γ 6 = 0 prevents this situation from happening.
For example, if
V (x) =
cosx sinx sinx − cosx
+ v(x)Id
with v smooth and at most quadratic, we have λ
1(x) = v(x) − 1 and λ
2(x) = v(x)+1:
classical trajectories for both modes, issued from the same point of the phase space, are equal.
1.3. Strategy of the proof of Theorem 1.9. The proof is more complicated than in the scalar case [4], due to the fact that the spectral projectors do not commute with the Laplace operator. From this perspective, a much finer geometric understanding is needed and we revisit [8, 7, 9, 13, 14] by adapting to our nonlinear context ideas contained therein.
Observe first that the function ϕ
εsatisfies (1.15) iε∂
tϕ
ε+ ε
22 ∂
x2ϕ
ε= T
ε(t, x)ϕ
ε+ Λ ε
3/2| ϕ
ε|
2CNϕ
ε, where
T
ε(t, x) = λ
1(x(t)) + λ
′1(x(t))(x − x(t)) + 1
2 λ
′′(x(t))(x − x(t))
2.
This term corresponds to the beginning of the Taylor expansion of λ
1about x(t).
Therefore, the function w
ε(t, x) = ψ
ε(t, x) − ϕ
ε(t, x)χ
1(t, x) satisfies w
|t=0ε= r
ε0and iε∂
tw
ε(t, x) + ε
22 ∂
x2w
ε(t, x) − V (x)w
ε(t, x) = ε N L g
ε(t, x) + ε L e
ε(t, x) where
N L g
ε= Λ ε
1/2ϕ
εχ
1+ w
ε2CN
(ϕ
εχ
1+ w
ε) − | ϕ
ε|
2ϕ
εχ
1, L e
ε= i∂
tχ
1ϕ
ε+ ε∂
xχ
1∂
xϕ
ε+ ε
2 ϕ
ε∂
x2χ
1+ ε
−1(λ
1(x) − T
ε) ϕ
εχ
1. Since ϕ
εis concentrated near x = x(t) at scale √
ε, we have (λ
1(x) − T
ε)ϕ
ε= O
ε
3/2e
Ctin L
2(R),
where we have used Theorem 1.7. The term L e
εa priori presents an O (1) con- tribution, which is an obstruction to infer that w
εis small by applying Gronwall Lemma. Observing that in view of the estimates on the classical flow (see (1.8))
ε∂
xϕ
ε= iξ(t)ϕ
ε+ O √ ε e
Ctin L
2( R ), we write,
L e
ε= i ∂
tχ
1+ ξ(t)∂
xχ
1ϕ
ε+ O √ ε e
Ctin L
2(R).
The choice of the time-dependent eigenvectors ensures that for all time, the O (1) contribution of L e
εis orthogonal to the first mode (the eigenspace associated with λ
1). Then, to get rid of these terms, we introduce a correction term to w
ε. We set
θ
ε(t, x) = w
ε(t, x) + εg
ε(t, x), g
ε(t, x) = X
26j6P
X
16ℓ6dj
g
εj,ℓ(t, x)χ
ℓj(x), where for j > 2 and for 1 6 ℓ 6 d
j, the function g
j,ℓε(t, x) solves the scalar Schr¨odinger equation
(1.16) iε∂
tg
εj,ℓ+ ε
22 ∂
x2g
j,ℓε− λ
j(x)g
εj,ℓ= ϕ
εr
j,ℓ; g
εj,ℓ|t=0= 0, where
(1.17) r
j,ℓ(t, x) = − i ∂
tχ
1(t, x) + ξ(t)∂
xχ
1(t, x) , χ
ℓj(x)
CN
.
The function θ
ε(t) then solves
(1.18)
iε∂
tθ
ε(t, x) + ε
22 ∂
x2θ
ε(t, x) = V (x)θ
ε(t, x) + εN L
ε(t, x) + εL
ε(t, x), θ
ε|t=0= r
ε0,
with
N L
ε= Λ ε
1/2| ϕ
εχ
1+ θ
ε− εg
ε|
2CN(ϕ
εχ
1+ θ
ε− εg
ε) − | ϕ
ε|
2ϕ
εχ
1, (1.19)
L
ε= L e
ε+
iε∂
t+ ε
22 ∂
x2− V (x)
g
ε(t, x) (1.20)
= O ( √
εe
Ct) + X
26j6P
X
16ℓ6dj
ε
22 ∂
x2, χ
ℓjg
εj,ℓwhere the O ( √
εe
Ct) holds in L
2. The proof of the theorem then follows from a precise control of the functions χ
ℓjand g
εj,ℓ, which is achieved in Sections 2 and 3, respectively. Then, the analysis of θ
εas ε goes to zero by an energy method is presented in Section 4.
2. The family of time-dependent eigenvectors
In this section we prove Proposition 1.8, recalling the construction of the eigen- vectors satisfying (1.14), and analyzing the behavior of their derivatives for large time. We follow the proof of [8]. More generally, we prove the following result which implies Proposition 1.8. We consider the Hamiltonian curves of
12| ξ |
2+ λ
j(x), that we denote by (x
j(t), ξ
j(t)).
Proposition 2.1. There exists a smooth orthonormal basis of C
Nχ
ℓj(t, x)
16ℓ6dj
16j6P
such that for all t, χ
ℓj(t, x)
16ℓ6dj
spans the eigenspace associated to λ
j, with χ
1(0, x) = χ(x) and for m ∈ { 1, · · · , d
j} ,
χ
mj(t, x), ∂
tχ
ℓj(t, x) + ξ
j(t)∂
xχ
ℓj(t, x)
CN
= 0.
Moreover, for ℓ ∈ { 1, . . . , d
j} , k, p ∈ N, there exists a constant C = C(p, k) such that ∂
tp∂
kxχ
ℓj(t, x)
CN6 C e
Cth x i
(k+p)(1+n0),
where n
0appears in (1.2).
Proof of Proposition 2.1. We consider a smooth basis of eigenvectors (χ
ℓj(0))
16ℓ6dj16j6P
such that χ
11(0) = χ. Then, we denote by Π
j(x) the smooth eigenprojector associ- ated with the eigenvalue λ
j(x) and define
K
j(x) = − i [Π
j(x), ∂
xΠ
j(x)] .
We set z = x − x
j(t) and we consider the Schr¨odinger type equation
(2.1) i∂
tY
jℓ(t, z) = ξ
j(t)K
j(z + x
j(t))Y
jℓ(t, z) ; Y
jℓ(0, z) = χ
ℓj(x
j(0) + z).
Let us prove that the vector Y
jℓ(t, z) is in the eigenspace of λ
j(x
j(t) + z). Indeed, the evolution of Z
jℓ(t, z) = (Id − Π
j(x
j(t) + z)) Y
jℓ(t, z) obeys to Z
jℓ(0, z) = 0 and
∂
tZ
jℓ(t, z) = − ξ
j(t)∂
xΠ
j(x
j(t) + z)Y
jℓ− ξ
j(t)(Id − Π
j(x
j(t) + z))[Π
j(x
j(t) + z), ∂
xΠ
j(x
j(t) + z)]Y
jℓ= − ξ
j(t)∂
xΠ
j(x
j(t) + z)(Id − Π
j(x
j(t) + z))Y
jℓ= − ξ
j(t)∂
xΠ
j(x
j(t) + z)Z
jℓwhere we have used ∂
xΠ
j= ∂
x(Π
2j) = Π
j∂
xΠ
j+ (∂
xΠ
j) Π
j, whence Π
j(∂
xΠ
j)Π
j= Π
j(Π
j∂
xΠ
j+ (∂
xΠ
j) Π
j) Π
j= 2Π
j(∂
xΠ
j)Π
j= 0.
Therefore, Z
jℓ(t) satisfies an equation of the form ∂
tZ
jℓ= A(t, z)Z
jℓ, which combined with Z
jℓ(0) = 0, implies Z
jℓ(t) = 0 for all t ∈ R: the vectors Y
jℓ(t, z) are eigenvectors of V (x
j(t) + z) for the eigenvalue λ
j(x
j(t) + z).
Besides, since ξ
j(t)K
j(z + x
j(t)) is self-adjoint, Y
jℓ(t, z) is normalized for all t, and the family (Y
jℓ)
16ℓ6djis orthonormal. We define χ
ℓj(t, x) by
(2.2) χ
ℓj(t, x) = Y
jℓ(t, x − x
j(t)) and we obtain an orthonormal basis of eigenvectors of V (x).
It remains to check that (1.14) holds. We have
∂
tχ
ℓj+ ξ
j(t)∂
xχ
ℓj= ∂
tY
jℓ(t, x − x
j(t))
= iξ
j(t)K
j(x)χ
ℓj= − ξ
j(t)[Π
j(x), ∂
xΠ
j(x)]χ
ℓj, whence
∂
tχ
ℓj+ ξ
j(t)∂
xχ
ℓj, χ
kjCN
= − ξ
j(t) [Π
j, ∂
xΠ
j]χ
ℓj, χ
kjCN
= − ξ
j(t) Π
j[Π
j, ∂
xΠ
j]Π
jχ
ℓj, χ
kjCN
since χ
ℓ/kj= Π
jχ
ℓ/kj. We then observe that Π
2j= Π
jimplies Π
j[Π
j, ∂
xΠ
j]Π
j= Π
2j∂
xΠ
jΠ
j− Π
j∂
xΠ
jΠ
2j= 0.
This concludes the first part of Proposition 1.8. It remains to study the behavior at infinity of the vectors χ
ℓj(t, x) and of their derivatives.
By the definition of χ
ℓj(t, x) in (2.2), it is enough to prove
| ∂
tp∂
xkY
jℓ(t, z) |
CN. e
Cth x
j(t) + z i
(p+k)(1+n0).
For this, we crucially use the estimates of Lemma C.2 and we argue by induction.
Let us first consider the case p = 1 and k = 0. By Lemma C.2, we have | K
j(x) | . h x i
1+n0, whence (2.1) gives
| ∂
tY
jℓ(t, x − x
j(t)) | . | ξ
j(t) || K
j(x) | . e
Cth x i
1+n0. Let us now suppose k > 1 and p = 0. We observe that ∂
zkY
jℓ(t, z) solves (2.3)
( i∂
t∂
zkY
jℓ(t, z) = − iξ
j(t)K
j(z + x
j(t))∂
zkY
jℓ(t, z) + f (t, z),
∂
zkY
j(0, z) = ∂
xkχ
ℓj(0, z + x
j(0)),
where
f (t, z) = X
06γ6k−1
c
γξ
j(t)∂
zγK
j(z + x
j(t))∂
zk−γY
jℓ(t, z), for some complex numbers c
γindependent of t and z. We obtain
∂
zkY
jℓ(t, z) = U
j(t, 0)∂
xkχ
ℓj(z + x
j(0)) + Z
t0
U
j(t, s)f (s, z)ds,
where U
j(t, s) denotes the unitary propagator associated to (2.1) (when the initial time is equal to s). We have by Lemma C.2
| ∂
zkY
jℓ(0, z) |
CN= ∂
xkχ
ℓj(0, z + x
j(0))
CN. h z + x
j(0) i
k(1+n0), therefore the induction assumption
∀ γ ∈ { 0, . . . , k − 1 } , | ∂
xγY
jℓ(t, z) |
CN. e
Cth x
j(t) + z i
γ(1+n0)implies, along with Lemma C.2,
| ∂
xkY
jℓ(t, z) |
CN. e
Cth x
j(t) + z i
k(1+n0).
We have obtained the estimate for p = 0, k ∈ N, and for p = 1, k = 0. Note that Equation (2.3) yields
∀ k ∈ N, | ∂
t∂
xkY
jℓ(t, z) |
CN. e
Cth x
j(t) + z i
(1+k)(1+n0),
and allows to prove the general estimate for time derivatives by an induction ar- gument which crucially uses the fact that we have an exponential control of the derivatives in time of ξ
j(t). This property follows by induction from (1.7), (1.8),
and the fact that λ
jis at most quadratic.
Before concluding this section, note that in view of the definition of the function r
j,ℓin (1.17), Proposition 1.8 gives the following corollary.
Corollary 2.2. For all p ∈ N and k ∈ N, there exists a constant C = C(p, k) such that, for x ∈ R, j ∈ { 1, · · · , P } and ℓ ∈ { 1, · · · , d
j} ,
∂
tp∂
kxr
j,ℓ(t, x) . e
Cth x i
(1+p+k)(1+n0). 3. Analysis of the correction terms
In this section, we will make use of the following norms defined for p ∈ N, k f k
Σpε= sup
α+β6p
| x |
αε
βf
(β)(x)
L2. We associate with this norm the functional space Σ
pεdefined by
Σ
pε= { f ∈ L
2(R
d), k f k
Σpε< ∞} .
In view of (1.8) and (1.13), for all p ∈ N, there exists c(p) such that (3.1) k ϕ
ε(t) k
Σpε. e
c(p)t, ∀ t > 0.
We can obviously take c(0) = 0 by conservation of the L
2-norm, but in general, the norm of ϕ
εin Σ
1εpotentially grows exponentially in time (see [3]). We denote by U
kε(t) the semi-group associated with the operator −
ε22∂
x2+ λ
k(x) and we observe that for p ∈ N, there exists a constant C(p) such that
(3.2) k U
kε(t) k
L(Σpε)6 C(p)e
C(p)|t|.
The following averaging lemma shows an asymptotic orthogonality property.
Lemma 3.1. For T > 0 and k 6 = j, there exists a constant C such that
∀ t ∈ [0, T ], ∀ p ∈ N , 1 iε
Z
t 0U
kε( − s)U
jε(s)ds
L
Σ(p+3)nε 0 +p+2,Σpε
6 Ce
Ct.
Proof. We first observe that (3.3) iε∂
tU
kε( − t)U
jε(t)
= U
kε( − t) (λ
j(x) − λ
k(x)) U
jε(t).
Indeed, if f ∈ L
2(R) and f
ε(t) = U
kε( − t)U
jε(t)f. We have iε∂
tf
ε(t, x) = −
− ε
22 ∂
x2+ λ
k(x)
f
ε(t) + U
kε( − t)
− ε
22 ∂
x2+ λ
j(x)
U
jε(t)f
= U
kε( − t) (λ
j(x) − λ
k(x)) U
jε(t)f
because U
kε( − t) commutes with −
ε22∂
x2+ λ
k(x). We use Equation (3.3) to perform an integration by parts:
U
kε( − t)U
jε(t) = U
kε( − t) (λ
j− λ
k)
−1U
kε(t)U
kε( − t) (λ
j− λ
k) U
jε(t)
= iε U
kε( − t) (λ
j− λ
k)
−1U
kε(t) ∂
tU
kε( − t)U
jε(t) . Therefore,
1 iε
Z
t 0U
kε( − s)U
jε(s)ds = h
U
kε( − s) (λ
j− λ
k)
−1U
jε(s) i
t 0− Z
t0
∂
sU
kε( − s) (λ
j− λ
k)
−1U
kε(s)
U
kε( − s)U
jε(s) ds.
Set
(3.4) γ
j,k= (λ
k− λ
j)
−1.
The behavior as x goes to infinity of these functions is studied in Appendix C (see Lemma C.1). It is proven there that for all β ∈ N,
∂
xβγ
j,k(x) . h x i
n0+|β|(1+n0).
Since the propagators U
kε(t) and U
jε(t) map continuously Σ
pεinto itself uniformly with respect to ε, we have
U
kε( − s)γ
j,kU
jε(s)
t 0L
Σ(p+1)nε 0 +p,Σpε
. C(p),
where in all this paragraph, C(p) denotes a generic constant depending only on the parameter p ∈ N. Besides, we observe that
∂
s(U
kε( − s)γ
j,kU
kε(s)) = 1
iε U
kε( − s)
− ε
22 ∂
x2+ λ
k, γ
j,kU
kε(s).
In view of 1 iε
− ε
22 ∂
x2+ λ
k, γ
j,k= 1 iε
− ε
22 ∂
x2, γ
j,k= iγ
j,k′(x)ε∂
x+ iεγ
j,k′′(x), and of
U
kε( − s)γ
′j,k(x)ε∂
xU
jε(s)
L
Σ(p+3)nε 0 +p+2,Σpε
+ U
kε( − s)γ
′′j,k(x)U
jε(s)
L
Σ(p+2)nε 0 +p+2,Σpε
. e
Cs,
which comes from (3.2) and Lemma C.1, we get
k ∂
s(U
kε( − s)γ
j,kU
kε(s)) k
L(Σp+2ε ,Σpε). e
Ct,
which concludes the proof.
We now prove the following proposition.
Proposition 3.2. For p ∈ N , there exists C(p) such that for all j > 2, and all ℓ ∈ { 1, . . . , d
j} ,
k g
j,ℓε(t) k
Σpε. e
C(p)t, ∀ t > 0, where g
j,ℓεis defined in (1.16).
Proof. We use Duhamel’s formula and write g
j,ℓε(t) = 1
iε Z
t0
U
jε(t − s) (ϕ
ε(s)r
j,ℓ(s)) ds.
Besides, if ˜ ϕ
εj,ℓ(t, x) = ϕ
ε(t, x)r
j,ℓ(t, x), then we have,
iε∂
t+ ε
22 ∂
x2− λ
1(x)
˜
ϕ
εj,ℓ= iε∂
tr
j,ℓϕ
ε+ r
j,ℓε
3/2| ϕ
ε|
2ϕ
ε+ ε
22
∂
x2, r
j,ℓ(t, x) ϕ
ε| {z }
=:εerε(t,x)
.
Therefore, we can write
˜
ϕ
εj,ℓ(t) = U
1ε(t) ˜ ϕ
εj,ℓ(0) − i Z
t0
U
1ε(t − s) e r
ε(s)ds, whence
g
j,ℓε(t) = 1 iε
Z
t 0U
jε(t − s)U
1ε(s)ds ϕ ˜
εj,ℓ(0) − 1 ε
Z
t 0Z
s 0U
jε(t − s)U
1ε(s − τ) r e
ε(τ)dτ ds
= 1 iε
Z
t 0U
jε(t − s)U
1ε(s)ds ϕ ˜
εj,ℓ(0) − Z
t0
1 ε
Z
t τU
jε(t − s)U
1ε(s − τ)ds
e r
ε(τ)dτ.
Lemma 3.1 yields
k g
j,ℓε(t) k
Σpε. e
Ct+ Z
t0
e
Cτke r
ε(τ) k
Σqεdτ,
with q = p + 2 + (p + 3)(1 + n
0). Let us now study e r
ε. We write e r
ε= e r
ε1+ e r
2εwith e
r
ε1(t, x) = i∂
tr
j,ℓϕ
ε+ ε 2
∂
x2, r
j,ℓ(t, x) ϕ
ε. In view of Corollary 2.2 and of (3.1), we have for all q ∈ N ,
ke r
ε1(t) k
Σqε(R). e
C(q)t. A very rough estimate yields
ke r
2ε(t) k
Σqε= k √
εr
j,ℓ| ϕ
ε|
2ϕ
εk
Σqε. √
ε k r
j,ℓh x i
qϕ
εk
Σqεkh ε∂
xi
qϕ
εk
2L∞. √
εe
Cth x i
q+(1+q)(n0+1)ϕ
εΣqε
kh ε∂
xi
qϕ
εk
2L∞,
where we have used Corollary 2.2. Now with (1.13) and (3.1), we conclude ke r
2ε(t) k
Σqε. e
Ct.
This completes the proof of Proposition 3.2.
4. Consistency
We now prove Theorem 1.9. We go back to Equation (1.18), that we recall:
iε∂
tθ
ε(t, x) + ε
22 ∂
x2θ
ε(t, x) = V (x)θ
ε(t, x) + εN L
ε(t, x) + εL
ε(t, x), θ
|t=0ε= r
ε0,
where N L
εand L
εare defined in (1.19) and (1.20), respectively. The standard L
2-estimate yields:
k θ
ε(t) k
L26 k r
ε0k
L2+ Z
t0
( k N L
ε(s) k
L2+ k L
ε(s) k
L2) ds.
In view of (1.20), Proposition 2.1 and Proposition 3.2, we have k L
ε(t) k
L2. √
εe
Ct. Besides, we observe
k N L
ε(t) k
L2. √
ε | ϕ
ε(t) |
2+ | θ
ε(t) |
2CN+ ε
2| g
ε(t) |
2CN(θ
ε(t) − εg
ε(t))
L2
. √
ε k ϕ
ε(t) k
2L∞+ k θ
ε(t) k
2L∞+ ε
2k g
ε(t) k
2L∞( k θ
ε(t) k
L2+ ε k g
ε(t) k
L2) . In view of (1.13), we have k ϕ
ε(t) k
L∞. ε
−1/4e
Ct. On the other hand, Proposi- tion 3.2 implies, in view of the Gagliardo-Nirenberg inequality
(4.1) k f k
L∞. ε
−1/2k f k
1/2L2k ε∂
xf k
1/2L2, the estimate
ε
2k g
ε(t) k
2L∞. εe
Ct.
Therefore, it is natural to perform a bootstrap argument assuming, say (4.2) k θ
ε(t) k
L∞6 ε
−1/4e
Ct.
Note that we fixed the value of the constant in factor of the right hand side equal to one. We did so because θ
ε, as an error term, is expected to be smaller than ϕ
ε(the approximate solution) in the limit ε → 0. As long as (4.2) holds, the L
2-estimate implies, in view of (1.5)
k θ
ε(t) k
L2. ε
κ+ Z
t0
√ εe
Cs+ e
Csk θ
ε(s) k
L2ds.
By Gronwall Lemma, we obtain
(4.3) k θ
ε(t) k
L26 C ε
κ+ √ ε
e
eCt.
It remains to check how long the bootstrap assumption (4.2) holds. For this, we use Gagliardo-Nirenberg inequality (4.1), and we look for a control of the norm of θ
ε(t) in Σ
1ε. Differentiating the system (1.18) with respect to x, we find
iε∂
t(ε∂
xθ
ε) + ε
22 ∂
x2(ε∂
xθ
ε) = V (x)ε∂
xθ
ε+ εV
′(x)θ
ε+ ε
2∂
xN L
ε+ ε
2∂
xL
ε, We observe that since V is at most quadratic, | V
′(x)θ
ε|
CN. h x i | θ
ε|
CN. Therefore, in order to obtain a closed system of estimates, we consider the equation satisfied by xθ
ε: multiply (1.18) by x,
iε∂
t(xθ
ε) + ε
22 ∂
x2(xθ
ε) = V (x)(xθ
ε) + ε
2∂
xθ
ε+ εxN L
ε+ εxL
ε.
By Proposition 3.2, we have
k xL
ε(t) k
L2+ k ε∂
xL
ε(t) k
L2. √ εe
Ct. Besides,
| xN L
ε(t, x) |
CN. | φ
ε(t, x) |
2+ | θ
ε(t, x) |
2CN+ ε
2| g
ε(t, x) |
2CN×
× ( | xθ
ε(t, x) |
CN+ ε | xg
ε(t, x) |
CN) ,
| ε∂
xN L
ε(t, x) |
CN. | φ
ε(t, x) |
2+ | θ
ε(t, x) |
2CN+ ε
2| g
ε(t, x) |
2CN| ε∂
xθ
ε(t, x) |
CN+ ε | φ
ε(t, x) |
2+ | θ
ε(t, x) |
2CN+ ε
2| g
ε(t, x) |
2CN| ε∂
xg
ε(t, x) |
CN+ | ε∂
xφ
ε(t, x) | × | φ
ε(t, x) | × | θ
ε(t, x) |
CN+ ε | φ
ε(t, x) |
2× | ∂
xχ
1(t, x) |
CN× | θ
ε(t, x) |
CN.
Arguing as before and using again (1.13), we obtain that under (1.2) we have k ε∂
xθ
ε(t) k
L2+ k xθ
ε(t) k
L2. ε
κ+ √
ε e
eCt. Gagliardo–Nirenberg inequality then implies
k θ
ε(t) k
L∞. ε
−1/2ε
κ+ √ ε
e
eCt. We infer that (4.2) holds (at least) as long as
ε
κ−1/2+ 1
e
eCt≪ ε
−1/4e
Ct, which is ensured provided that t 6 Cloglog
1ε, for some suitable constant C, since κ > 1/4. This concludes the bootstrap argument: we infer
sup
|t|6Cloglog
(
1ε)
( k θ
ε(t) k
L2+ k xθ
ε(t) k
L2+ k ε∂
xθ
ε(t) k
L2) −→
ε→00.
Theorem 1.9 then follows from the above asymptotics, together with the relation θ
ε= w
ε+ εg
ε, and Proposition 3.2.
Remark 4.1. In the case where V
ε= D + εW , as in Remark 1.10, the proof can be adapted, in order to reproduce the argument given in [4]. The main point to notice is that (local in time) Strichartz estimates are available for the propagator associated to −
ε22∂
x2+ D(x), thanks to [6]. Then in the presence of the power ε in front of W , the potential εW can be considered as a source term in the error estimates: the factor ε is crucial to avoid a singular power of ε due to the presence of ε in front of the time derivative in (1.18). The proof in [4, Section 6] for the cubic, one-dimensional Schr¨odinger equation can be reproduced: another bootstrap argument can be invoked, which does not involve Gagliardo–Nirenberg inequalities, since a useful a priori estimate for the envelope u is available.
5. Superposition
As explained in the introduction, the only difficulty in the proof of Theorem 1.11 is to treat a nonlinear interaction term. Indeed, we set
w
ε= ψ
ε− ϕ
ε1χ
1− ϕ
ε2χ
2+ εg
εwhere g
εis the sum of two correction terms, similar to the one introduced in § 1.3.
More precisely, set p(1) = 1, and p(2) = 1 if ˜ λ
1= ˜ λ
2, p(2) = 2 otherwise. Define
g
ε= g
1ε+ g
ε2, with
g
1ε= X
16j6P, j6=p(1)
X
16ℓ6dj
g
j,1,ℓε(t, x)χ
ℓj(t, x), g
2ε= X
16j6P, j6=p(2)
X
16ℓ6dj
g
j,2,ℓε(t, x)χ
ℓj(t, x),
where for k = { 1, 2 } , j 6 = p(k) and 1 6 ℓ 6 d
j, the function g
εj,k,ℓ(t, x) solves the scalar Schr¨odinger equation
(5.1) iε∂
tg
j,k,ℓε+ ε
22 ∂
x2g
εj,k,ℓ− λ
j(x)g
εj,k,ℓ= ϕ
εr
j,k,ℓ; g
j,k,ℓ|t=0ε= 0, where
(5.2) r
j,k,ℓ(t, x) = − i ∂
tχ
k(t, x) + ξ
p(k)(t)∂
xχ
k(t, x) , χ
ℓj(t, x)
CN
. The function w
ε(t) then solves
iε∂
tw
ε+ ε
22 ∂
x2w
ε= V (x)w
ε+ εN L
ε+ εL
ε; w
ε|t=0= 0, with
L
ε= O ( √
εe
Ct) + X
k=1,2
X
16j6P j6=p(k)
X
16ℓ6dj
ε
22 ∂
x2, χ
ℓjg
j,k,ℓε= O ( √ εe
Ct).
Here, the O ( √
εe
Ct) holds in Σ
1ε, from Proposition 3.2. Besides, N L
ε= √
ε w
ε+ ϕ
ε1χ
1+ ϕ
ε2χ
2+ εg
ε2
w
ε+ ϕ
ε1χ
1+ ϕ
ε2χ
2+ εg
ε− | ϕ
ε1|
2ϕ
ε1χ
1− | ϕ
ε2|
2ϕ
ε2χ
2Adding and subtracting the term √
ε | ϕ
ε1χ
1+ ϕ
ε2χ
2|
2(ϕ
ε1χ
1+ ϕ
ε2χ
2), we have
| N L
ε| 6 N
Sε+ N
Iε, where we have the pointwise estimates
N
Iε. √
ε | ϕ
ε1|
2| ϕ
ε2| + | ϕ
ε2|
2| ϕ
ε1| , N
Sε. √
ε | ϕ
ε1|
2+ | ϕ
ε2|
2+ | w
ε|
2+ ε
2| g
ε|
2( | w
ε| + ε | g
ε| ) .
The semilinear term N
Sεcan be treated exactly in the same manner as in Section 4.
It remains to analyze Z
t0
k N L
εI(s) k
Σ1εds. We observe
√ ε Z
t0
| ϕ
ε1(s) |
2ϕ
ε2(s)
L2
ds = Z
t0
u
1s, y − x
1(s) − x
2(s)
√ ε
2
u
2(s, y)
L2ds,
and we note that the contribution of | ϕ
ε1|
2ϕ
ε2and that of | ϕ
ε2|
2ϕ
ε1play the same role. Also, we leave out the other terms which are needed in view of a Σ
1εestimate, since they create no trouble. Arguing as in [4, Lemma 6.1], we obtain:
Lemma 5.1. Let T ∈ R, 0 < γ < 1/2 and
I
ε(T ) = { t ∈ [0, T ], | x
1(t) − x
2(t) | 6 ε
γ} .
Then, for all k ∈ N, there exists a constant C
ksuch that Z
T0
k N L
εI(t) k
Σ1εdt . (M
k+2(T ))
3T ε
k(1/2−γ)+ | I
ε(T ) | e
CkT, with
M
k(T ) = sup
k h x i
α∂
xβu
jk
L∞([0,T],L2(R)); j ∈ { 1, 2 } , α + β 6 k . In view of this lemma and of Equation (1.12), we obtain
Z
T0
k N L
εI(t) k
Σ1εdt . e
CTT ε
k(1/2−γ)+ | I
ε(T ) | ,
and the next lemma yields the conclusion.
Lemma 5.2. Set
Γ = inf
x∈R
